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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">x0</font>, <font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</font>, <font color="Maroon" title="c7">x3</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st ( for <font color="Olive" title="b2">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">x</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">c</font> ) &amp; <font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font> <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_different</a>  holds <br/><span class="p1"><a href="diff_1.html#K14" title="DIFF_1:func.14">[!</a><span class="default"><font color="Olive" title="b1">f</font>,<font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="diff_1.html#K14" title="DIFF_1:func.14">!]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c8">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">c</font> ) &amp; <font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font> <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_different</a>  implies <span class="p1"><a href="diff_1.html#K14" title="DIFF_1:func.14">[!</a><span class="default"><font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="diff_1.html#K14" title="DIFF_1:func.14">!]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>















<span class="kw">assume </span><a NAME="E1:31"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font> <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_different</a>  or <span class="p1"><a href="diff_1.html#K14" title="DIFF_1:func.14">[!</a><span class="default"><font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="diff_1.html#K14" title="DIFF_1:func.14">!]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:31"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font> <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_different</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="diff_1.html#K14" title="DIFF_1:func.14">[!</a><span class="default"><font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="diff_1.html#K14" title="DIFF_1:func.14">!]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E3:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D6" title="ZFMISC_1:def.6">ZFMISC_1:def 6</a></span>;<br/>
<a NAME="E4:31"/>
( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">x3</font> &amp; <font color="Maroon" title="c6">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">x3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#D6" title="ZFMISC_1:def.6">ZFMISC_1:def 6</a></span>;<br/>
<a NAME="E5:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a></span>;<br/>
<a NAME="E6:31"/>
( <font color="Maroon" title="c4">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x1</font> &amp; <font color="Maroon" title="c4">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#D6" title="ZFMISC_1:def.6">ZFMISC_1:def 6</a></span>;<br/>
<a NAME="E7:31"/><span class="kw">then </span>
<font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="p1"><a href="diff_1.html#K14" title="DIFF_1:func.14">[!</a><span class="default"><font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="diff_1.html#K14" title="DIFF_1:func.14">!]</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2"><a href="diff_1.html#K13" title="DIFF_1:func.13">[!</a><span class="default"><font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="diff_1.html#K13" title="DIFF_1:func.13">!]</a></span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">x3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="diff_2.html#T28" target="_self" title="DIFF_2:th.28">Th28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">x3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="diff_2.html#T28" target="_self" title="DIFF_2:th.28">Th28</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:31"/>
<span class="p1"><a href="diff_1.html#K14" title="DIFF_1:func.14">[!</a><span class="default"><font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="diff_1.html#K14" title="DIFF_1:func.14">!]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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